If $H$ and $K$ are conjugate in $G$, are their normalizers equal, or just isomorphic?

Question

I've been working on a problem that states:

"Suppose $H$ is conjugate to $K$ in $G$ . Determine the relationship between $N_G(H)$ and $N_G(K)$".

I thought the relationship was isomorphic and so I constructed the isomorphism $\phi(n) = gng^{-1}$ and proved it was, in fact, an isomorphism.

However, twice it's been mentioned to me (without proof) that the relationship is actually equality. I've been unable to prove that they are equal.

So, my question is, are the normalizers just isomorphic or are they actually equal? And if they are equal, could someone provide a hint?

Thank you!

Answer 1

Let $G$ be the nonabelian group of order 6 (permutations of 3 objects), and let $H$ and $K$ be two of its subgroups of order 2. Then $H$ and $K$ are conjugate, and each is its own normalizer.